论文标题
最高度三的图形尺寸尺寸
Size-Ramsey numbers of graphs with maximum degree three
论文作者
论文摘要
图$ h $的尺寸ramsey number $ \ hat {r}（h）$是最小的边缘a（主机）图$ g $可以具有的，因此对于$ g $的任何红色/蓝色颜色，都有$ g $中的$ h $的单色副本。最近，Conlon，Nenadov和Trujić表明，如果$ h $是$ n $顶点的图表和最高学位的最高学位，则是$ \ hat {r}（h）= o（n^{8/5}）$，在$ n^{5/3 + o（1）$ n^$ n^的上限上得到改善，可用于Kohayakawa，Rourtar，rorter，sz and sch and sch。在本文中，我们表明$ \ hat {r}（h）\ leq n^{3/2+o（1）} $。虽然先前使用的主机图是香草二项式随机图，但我们使用新型的宿主图构造证明了结果。我们的界限击中了现有方法的自然障碍。
The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Trujić showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}(H) = O(n^{8/5})$, improving upon the upper bound of $n^{5/3 + o(1)}$ by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper we show that $\hat{r}(H)\leq n^{3/2+o(1)}$. While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.
